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Theorem List for Metamath Proof Explorer - 25701-25800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnbgracnvfv 25701 Applying the edge function on the converse edge function applied on a pair of a vertex and one of its neighbors is this pair. (Contributed by Alexander van der Vekens, 18-Dec-2017.)
((𝑉 USGrph 𝐸𝑁 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈)) → (𝐸‘(𝐸‘{𝑈, 𝑁})) = {𝑈, 𝑁})
 
Theoremnbgraf1olem1 25702* Lemma 1 for nbgraf1o 25708. For each neighbor of a vertex there is exactly one index for the edge between the vertex and its neighbor. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑈)    &   𝐼 = {𝑖 ∈ dom 𝐸𝑈 ∈ (𝐸𝑖)}    &   𝐹 = (𝑛𝑁 ↦ (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛}))       (((𝑉 USGrph 𝐸𝑈𝑉) ∧ 𝑀𝑁) → ∃!𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑀})
 
Theoremnbgraf1olem2 25703* Lemma 2 for nbgraf1o 25708. The mapping of neighbors to edge indices is a function. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑈)    &   𝐼 = {𝑖 ∈ dom 𝐸𝑈 ∈ (𝐸𝑖)}    &   𝐹 = (𝑛𝑁 ↦ (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛}))       ((𝑉 USGrph 𝐸𝑈𝑉) → 𝐹:𝑁𝐼)
 
Theoremnbgraf1olem3 25704* Lemma 3 for nbgraf1o 25708. The restricted iota of an edge is the function value of the converse applied to the edge. (Contributed by Alexander van der Vekens, 18-Dec-2017.)
𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑈)    &   𝐼 = {𝑖 ∈ dom 𝐸𝑈 ∈ (𝐸𝑖)}    &   𝐹 = (𝑛𝑁 ↦ (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛}))       ((𝑉 USGrph 𝐸𝑈𝑉𝑀𝑁) → (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑀}) = (𝐸‘{𝑈, 𝑀}))
 
Theoremnbgraf1olem4 25705* Lemma 4 for nbgraf1o 25708. The mapping of neighbors to edge indices applied on a neighbor is the function value of the converse applied on the edge between the vertex and this neighbor. (Contributed by Alexander van der Vekens, 18-Dec-2017.)
𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑈)    &   𝐼 = {𝑖 ∈ dom 𝐸𝑈 ∈ (𝐸𝑖)}    &   𝐹 = (𝑛𝑁 ↦ (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛}))       ((𝑉 USGrph 𝐸𝑈𝑉𝑀𝑁) → (𝐹𝑀) = (𝐸‘{𝑈, 𝑀}))
 
Theoremnbgraf1olem5 25706* Lemma 5 for nbgraf1o 25708. The mapping of neighbors to edge indices is a one-to-one onto function. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑈)    &   𝐼 = {𝑖 ∈ dom 𝐸𝑈 ∈ (𝐸𝑖)}    &   𝐹 = (𝑛𝑁 ↦ (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛}))       ((𝑉 USGrph 𝐸𝑈𝑉) → 𝐹:𝑁1-1-onto𝐼)
 
Theoremnbgraf1o0 25707* The set of neighbors of a vertex is isomorphic to the set of indices of edges containing the vertex. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑈)    &   𝐼 = {𝑖 ∈ dom 𝐸𝑈 ∈ (𝐸𝑖)}       ((𝑉 USGrph 𝐸𝑈𝑉) → ∃𝑓 𝑓:𝑁1-1-onto𝐼)
 
Theoremnbgraf1o 25708* The set of neighbors of a vertex is isomorphic to the set of indices of edges containing the vertex. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
((𝑉 USGrph 𝐸𝑈𝑉) → ∃𝑓 𝑓:(⟨𝑉, 𝐸⟩ Neighbors 𝑈)–1-1-onto→{𝑖 ∈ dom 𝐸𝑈 ∈ (𝐸𝑖)})
 
Theoremnbusgrafi 25709 The class of neighbors of a vertex in a finite graph is a finite set. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
((𝑉 USGrph 𝐸𝑁𝑉𝐸 ∈ Fin) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∈ Fin)
 
Theoremnbfiusgrafi 25710 The class of neighbors of a vertex in a finite graph is a finite set. (Contributed by Alexander van der Vekens, 7-Mar-2018.)
((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∈ Fin)
 
Theoremedgusgranbfin 25711* The number of neighbors of a vertex in a graph is finite, if and only if the number of edges having this vertex as endpoint is finite. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
((𝑉 USGrph 𝐸𝑈𝑉) → ((⟨𝑉, 𝐸⟩ Neighbors 𝑈) ∈ Fin ↔ {𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)} ∈ Fin))
 
Theoremnb3graprlem1 25712 Lemma 1 for nb3grapr 25714. (Contributed by Alexander van der Vekens, 15-Oct-2017.)
(((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → ((⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶} ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)))
 
Theoremnb3graprlem2 25713* Lemma 2 for nb3grapr 25714. (Contributed by Alexander van der Vekens, 17-Oct-2017.)
(((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶} ↔ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣})(⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝑣, 𝑤}))
 
Theoremnb3grapr 25714* The neighbors of a vertex in a graph with three elements are an unordered pair of the other vertices if and only if all vertices are connected with each other. (Contributed by Alexander van der Vekens, 18-Oct-2017.)
(((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ↔ ∀𝑥𝑉𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝑥) = {𝑦, 𝑧}))
 
Theoremnb3grapr2 25715 The neighbors of a vertex in a graph with three elements are an unordered pair of the other vertices if and only if all vertices are connected with each other. (Contributed by Alexander van der Vekens, 18-Oct-2017.)
(((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ↔ ((⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶} ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐵) = {𝐴, 𝐶} ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐶) = {𝐴, 𝐵})))
 
Theoremnb3gra2nb 25716 If the neighbors of two vertices in a graph with three elements are an unordered pair of the other vertices, the neighbors of all three vertices are an unordered pair of the other vertices. (Contributed by Alexander van der Vekens, 18-Oct-2017.)
(((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → (((⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶} ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐵) = {𝐴, 𝐶}) ↔ ((⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶} ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐵) = {𝐴, 𝐶} ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐶) = {𝐴, 𝐵})))
 
16.1.4.2  Complete graphs
 
Theoremiscusgra 25717* The property of being a complete (undirected simple) graph. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
((𝑉𝑋𝐸𝑌) → (𝑉 ComplUSGrph 𝐸 ↔ (𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸)))
 
Theoremiscusgra0 25718* The property of being a complete (undirected simple) graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
(𝑉 ComplUSGrph 𝐸 → (𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸))
 
Theoremcusisusgra 25719 A complete (undirected simple) graph is an undirected simple graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
(𝑉 ComplUSGrph 𝐸𝑉 USGrph 𝐸)
 
Theoremcusgrarn 25720* In a complete simple graph, the range of the edge function consists of all the pairs with different vertices. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
(𝑉 ComplUSGrph 𝐸 → ran 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
 
Theoremcusgra0v 25721 A graph with no vertices (and therefore no edges) is complete. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
∅ ComplUSGrph ∅
 
Theoremcusgra1v 25722 A graph with one vertex (and therefore no edges) is complete. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
{𝐴} ComplUSGrph ∅
 
Theoremcusgra2v 25723 A graph with two (different) vertices is complete if and only if there is an edge between these two vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} USGrph 𝐸 → ({𝐴, 𝐵} ComplUSGrph 𝐸 ↔ {𝐴, 𝐵} ∈ ran 𝐸)))
 
Theoremnbcusgra 25724 In a complete (undirected simple) graph, each vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
((𝑉 ComplUSGrph 𝐸𝑁𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = (𝑉 ∖ {𝑁}))
 
Theoremcusgra3v 25725 A graph with three (different) vertices is complete if and only if there is an edge between each of these three vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
𝑉 = {𝐴, 𝐵, 𝐶}       (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝑉 USGrph 𝐸 ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝑉 ComplUSGrph 𝐸 ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)))
 
Theoremcusgra3vnbpr 25726* The neighbors of a vertex in a graph with three elements are unordered pairs of the other vertices if and only if the graph is complete. (Contributed by Alexander van der Vekens, 18-Oct-2017.)
𝑉 = {𝐴, 𝐵, 𝐶}       (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝑉 USGrph 𝐸 ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝑉 ComplUSGrph 𝐸 ↔ ∀𝑥𝑉𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝑥) = {𝑦, 𝑧}))
 
Theoremcusgraexilem1 25727* Lemma 1 for cusgraexi 25729. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}       (𝑉𝑊 → ( I ↾ 𝑃) ∈ V)
 
Theoremcusgraexilem2 25728* Lemma 2 for cusgraexi 25729. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}       (𝑉𝑊𝑉 USGrph ( I ↾ 𝑃))
 
Theoremcusgraexi 25729* For each set the identity function restricted to the set of pairs of elements from the given set is an edge function, so that the given set together with this edge function is a complete graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}       (𝑉𝑊𝑉 ComplUSGrph ( I ↾ 𝑃))
 
Theoremcusgraexg 25730* For each set there is an edge function so that the set together with this edge function is a complete graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
(𝑉𝑊 → ∃𝑒 𝑉 ComplUSGrph 𝑒)
 
Theoremcusgrasizeindb0 25731 Base case of the induction in cusgrasize 25738. The size of a complete simple graph with 0 vertices is 0=((0-1)*0)/2. (Contributed by Alexander van der Vekens, 2-Jan-2018.)
((𝑉 ComplUSGrph 𝐸 ∧ (#‘𝑉) = 0) → (#‘𝐸) = ((#‘𝑉)C2))
 
Theoremcusgrasizeindb1 25732 Base case of the induction in cusgrasize 25738. The size of a complete simple graph with 1 vertex is 0=((1-1)*1)/2. (Contributed by Alexander van der Vekens, 2-Jan-2018.)
((𝑉 ComplUSGrph 𝐸 ∧ (#‘𝑉) = 1) → (#‘𝐸) = ((#‘𝑉)C2))
 
Theoremcusgrares 25733* Restricting a complete simple graph. (Contributed by Alexander van der Vekens, 2-Jan-2018.)
𝐹 = (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})       ((𝑉 ComplUSGrph 𝐸𝑁𝑉) → (𝑉 ∖ {𝑁}) ComplUSGrph 𝐹)
 
Theoremcusgrasizeindslem1 25734* Lemma 1 for cusgrasizeinds 25736. (Contributed by Alexander van der Vekens, 11-Jan-2018.)
𝐹 = (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})       (dom 𝐹 ∩ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) = ∅
 
Theoremcusgrasizeindslem2 25735* Lemma 2 for cusgrasizeinds 25736. (Contributed by Alexander van der Vekens, 11-Jan-2018.)
𝐹 = (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})       ((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) = ((#‘𝑉) − 1))
 
Theoremcusgrasizeinds 25736* Part 1 of induction step in cusgrasize 25738. The size of a complete simple graph with 𝑛 vertices is (𝑛 − 1) plus the size of the complete graph reduced by one vertex. (Contributed by Alexander van der Vekens, 11-Jan-2018.)
𝐹 = (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})       ((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (#‘𝐸) = (((#‘𝑉) − 1) + (#‘𝐹)))
 
Theoremcusgrasize2inds 25737* Induction step in cusgrasize 25738. If the size of the complete graph with 𝑛 vertices reduced by one vertex is "(𝑛 − 1) choose 2", the size of the complete graph with 𝑛 vertices is "𝑛 choose 2". (Contributed by Alexander van der Vekens, 11-Jan-2018.)
𝐹 = (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})       (𝑌 ∈ ℕ0 → ((𝑉 ComplUSGrph 𝐸 ∧ (#‘𝑉) = 𝑌𝑁𝑉) → ((#‘𝐹) = ((#‘(𝑉 ∖ {𝑁}))C2) → (#‘𝐸) = ((#‘𝑉)C2))))
 
Theoremcusgrasize 25738 The size of a finite complete simple graph with 𝑛 vertices (𝑛 ∈ ℕ0) is (𝑛C2) ("𝑛 choose 2") resp. (((𝑛 − 1)∗𝑛) / 2). (Contributed by Alexander van der Vekens, 11-Jan-2018.)
((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin) → (#‘𝐸) = ((#‘𝑉)C2))
 
Theoremcusgrafilem1 25739* Lemma 1 for cusgrafi 25742. (Contributed by Alexander van der Vekens, 13-Jan-2018.)
𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁})}       ((𝑉 ComplUSGrph 𝐸𝑁𝑉) → 𝑃 ⊆ ran 𝐸)
 
Theoremcusgrafilem2 25740* Lemma 2 for cusgrafi 25742. (Contributed by Alexander van der Vekens, 13-Jan-2018.)
𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁})}    &   𝐹 = (𝑥 ∈ (𝑉 ∖ {𝑁}) ↦ {𝑥, 𝑁})       ((𝑉𝑊𝑁𝑉) → 𝐹:(𝑉 ∖ {𝑁})–1-1-onto𝑃)
 
Theoremcusgrafilem3 25741* Lemma 3 for cusgrafi 25742. (Contributed by Alexander van der Vekens, 13-Jan-2018.)
𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁})}    &   𝐹 = (𝑥 ∈ (𝑉 ∖ {𝑁}) ↦ {𝑥, 𝑁})       ((𝑉𝑊𝑁𝑉) → (¬ 𝑉 ∈ Fin → ¬ 𝑃 ∈ Fin))
 
Theoremcusgrafi 25742 If the size of a complete simple graph is finite, then also its order is finite. (Contributed by Alexander van der Vekens, 13-Jan-2018.)
((𝑉 ComplUSGrph 𝐸𝐸 ∈ Fin) → 𝑉 ∈ Fin)
 
Theoremusgrasscusgra 25743* An undirected simple graph is a subgraph of a complete simple graph. (Contributed by Alexander van der Vekens, 11-Jan-2018.)
((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → ∀𝑒 ∈ ran 𝐸𝑓 ∈ ran 𝐹 𝑒 = 𝑓)
 
Theoremsizeusglecusglem1 25744 Lemma 1 for sizeusglecusg 25746. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → ( I ↾ ran 𝐸):ran 𝐸1-1→ran 𝐹)
 
Theoremsizeusglecusglem2 25745 Lemma 2 for sizeusglecusg 25746. (Contributed by Alexander van der Vekens, 13-Jan-2018.)
((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹𝐹 ∈ Fin) → 𝐸 ∈ Fin)
 
Theoremsizeusglecusg 25746 The size of an undirected simple graph with 𝑛 vertices is at most the size of a complete simple graph with 𝑛 vertices (𝑛 may be infinite). (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Proof shortened by AV, 4-May-2021.)
((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → (#‘𝐸) ≤ (#‘𝐹))
 
Theoremusgramaxsize 25747 The maximum size of an undirected simple graph with 𝑛 vertices (𝑛 ∈ ℕ0) is (((𝑛 − 1)∗𝑛) / 2). (Contributed by Alexander van der Vekens, 13-Jan-2018.)
((𝑉 USGrph 𝐸𝑉 ∈ Fin) → (#‘𝐸) ≤ ((#‘𝑉)C2))
 
16.1.4.3  Universal vertices
 
Theoremisuvtx 25748* The set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
((𝑉𝑋𝐸𝑌) → (𝑉 UnivVertex 𝐸) = {𝑛𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸})
 
Theoremuvtxel 25749* A universal vertex, i.e. an element of the set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
((𝑉𝑋𝐸𝑌) → (𝑁 ∈ (𝑉 UnivVertex 𝐸) ↔ (𝑁𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁}){𝑘, 𝑁} ∈ ran 𝐸)))
 
Theoremuvtxisvtx 25750 A universal vertex is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
(𝑁 ∈ (𝑉 UnivVertex 𝐸) → 𝑁𝑉)
 
Theoremuvtx0 25751 There is no universal vertex if there is no vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
(∅ UnivVertex 𝐸) = ∅
 
Theoremuvtx01vtx 25752* If a graph/class has no edges, it has universal vertices if and only if it has exactly one vertex. This theorem could have been stated ((𝑉 UnivVertex ∅) ≠ ∅ ↔ (#‘𝑉) = 1), but a lot of auxiliary theorems would have been needed. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
((𝑉 UnivVertex ∅) ≠ ∅ ↔ ∃𝑥 𝑉 = {𝑥})
 
Theoremuvtxnbgra 25753 A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017.)
((𝑉 USGrph 𝐸𝑁 ∈ (𝑉 UnivVertex 𝐸)) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = (𝑉 ∖ {𝑁}))
 
Theoremuvtxnm1nbgra 25754 A universal vertex has 𝑛 − 1 neighbors in a graph with 𝑛 vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.)
((𝑉 USGrph 𝐸𝑉 ∈ Fin) → (𝑁 ∈ (𝑉 UnivVertex 𝐸) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1)))
 
Theoremuvtxnbgravtx 25755* A universal vertex is neighbor of all other vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.)
((𝑉 USGrph 𝐸𝑁 ∈ (𝑉 UnivVertex 𝐸)) → ∀𝑣 ∈ (𝑉 ∖ {𝑁})𝑁 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑣))
 
Theoremcusgrauvtxb 25756 An undirected simple graph is complete if and only if each vertex is a universal vertex. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by Alexander van der Vekens, 18-Jan-2018.)
(𝑉 USGrph 𝐸 → (𝑉 ComplUSGrph 𝐸 ↔ (𝑉 UnivVertex 𝐸) = 𝑉))
 
Theoremuvtxnb 25757 A vertex in a undirected simple graph is universal iff all the other vertices are its neighbors. (Contributed by Alexander van der Vekens, 13-Jul-2018.)
((𝑉 USGrph 𝐸𝑁𝑉) → (𝑁 ∈ (𝑉 UnivVertex 𝐸) ↔ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = (𝑉 ∖ {𝑁})))
 
16.1.5  Walks, paths and cycles
 
Syntaxcwalk 25758 Extend class notation with Walks (of a graph).
class Walks
 
Syntaxctrail 25759 Extend class notation with Trails (of a graph).
class Trails
 
Syntaxcpath 25760 Extend class notation with Paths (of a graph).
class Paths
 
Syntaxcspath 25761 Extend class notation with Simple Paths (of a graph).
class SPaths
 
Syntaxcwlkon 25762 Extend class notation with Walks between two vertices (within a graph).
class WalkOn
 
Syntaxctrlon 25763 Extend class notation with Trails between two vertices (within a graph).
class TrailOn
 
Syntaxcpthon 25764 Extend class notation with Paths between two vertices (within a graph).
class PathOn
 
Syntaxcspthon 25765 Extend class notation with simple paths between two vertices (within a graph).
class SPathOn
 
Syntaxccrct 25766 Extend class notation with Circuits (of a graph).
class Circuits
 
Syntaxccycl 25767 Extend class notation with Cycles (of a graph).
class Cycles
 
Definitiondf-wlk 25768* Define the set of all Walks (in an undirected graph).

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A walk of length k in a graph is an alternating sequence of vertices and edges, v0 , e0 , v1 , e1 , v2 , ... , v(k-1) , e(k-1) , v(k) which begins and ends with vertices. If the graph is undirected, then the endpoints of e(i) are v(i) and v(i+1)."

According to Bollobas: " A walk W in a graph is an alternating sequence of vertices and edges x0 , e1 , x1 , e2 , ... , e(l) , x(l) where e(i) = x(i-1)x(i), 0<i<=l.", see Definition of [Bollobas] p. 4.

Therefore, a walk can be represented by two mappings f from { 1 , ... , n } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices. So the walk is represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)

Walks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝑒𝑝:(0...(#‘𝑓))⟶𝑣 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})})
 
Definitiondf-trail 25769* Define the set of all Trails (in an undirected graph).

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A trail is a walk in which all edges are distinct.

According to Bollobas: "... walk is called a trail if all its edges are distinct.", see Definition of [Bollobas] p. 5.

Therefore, a trail can be represented by an injective mapping f from { 1 , ... , n } and a mapping p from { 0 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the trail is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)

Trails = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ Fun 𝑓)})
 
Definitiondf-pth 25770* Define the set of all Paths (in an undirected graph).

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A path is a trail in which all vertices (except possibly the first and last) are distinct. ... use the term simple path to refer to a path which contains no repeated vertices."

According to Bollobas: "... a path is a walk with distinct vertices.", see Notation of [Bollobas] p. 5. (A walk with distinct vertices is actually a simple path, see wlkdvspth 25870).

Therefore, a path can be represented by an injective mapping f from { 1 , ... , n } and a mapping p from { 0 , ... , n }, which is injective restricted to the set { 1 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the path is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)

Paths = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)})
 
Definitiondf-spth 25771* Define the set of all Simple Paths (in an undirected graph).

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A path is a trail in which all vertices (except possibly the first and last) are distinct. ... use the term simple path to refer to a path which contains no repeated vertices."

Therefore, a simple path can be represented by an injective mapping f from { 1 , ... , n } and an injective mapping p from { 0 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the simple path is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). (Contributed by Alexander van der Vekens, 20-Oct-2017.)

SPaths = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ Fun 𝑝)})
 
Definitiondf-crct 25772* Define the set of all circuits (in an undirected graph).

According to Wikipedia ("Cycle (graph theory)", https://en.wikipedia.org/wiki/Cycle_(graph_theory), 3-Oct-2017): "A circuit can be a closed walk allowing repetitions of vertices but not edges;"; according to Wikipedia ("Glossary of graph theory terms", https://en.wikipedia.org/wiki/Glossary_of_graph_theory_terms, 3-Oct-2017): "A circuit may refer to ... a trail (a closed tour without repeated edges), ...".

Following Bollobas ("A trail whose endvertices coincide (a closed trail) is called a circuit.", see Definition of [Bollobas] p. 5.), a circuit is a closed trail without repeated edges. So the circuit is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n)=p(0). (Contributed by Alexander van der Vekens, 3-Oct-2017.)

Circuits = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})
 
Definitiondf-cycl 25773* Define the set of all (simple) cycles (in an undirected graph).

According to Wikipedia ("Cycle (graph theory)", https://en.wikipedia.org/wiki/Cycle_(graph_theory), 3-Oct-2017): "A simple cycle may be defined either as a closed walk with no repetitions of vertices and edges allowed, other than the repetition of the starting and ending vertex,"

According to Bollobas: "If a walk W = x0 x1 ... x(l) is such that l >= 3, x0=x(l), and the vertices x(i), 0 < i < l, are distinct from each other and x0, then W is said to be a cycle.", see Definition of [Bollobas] p. 5.

However, since a walk consisting of distinct vertices (except the first and the last vertex) is a path, a cycle can be defined as path whose first and last vertices coincide. So a cycle is represented by the following sequence: p(0) e(f(1)) p(1) ... p(n-1) e(f(n)) p(n)=p(0). (Contributed by Alexander van der Vekens, 3-Oct-2017.)

Cycles = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Paths 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})
 
Definitiondf-wlkon 25774* Define the collection of walks with particular endpoints (in an un- directed graph). This corresponds to the "x0-x(l)-walks", see Definition in [Bollobas] p. 5. (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)
WalkOn = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎𝑣, 𝑏𝑣 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏)}))
 
Definitiondf-trlon 25775* Define the collection of trails with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)
TrailOn = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎𝑣, 𝑏𝑣 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝𝑓(𝑣 Trails 𝑒)𝑝)}))
 
Definitiondf-pthon 25776* Define the collection of paths with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)
PathOn = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎𝑣, 𝑏𝑣 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝𝑓(𝑣 Paths 𝑒)𝑝)}))
 
Definitiondf-spthon 25777* Define the collection of simple paths with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.)
SPathOn = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎𝑣, 𝑏𝑣 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝𝑓(𝑣 SPaths 𝑒)𝑝)}))
 
16.1.5.1  Walks and trails
 
Theoremrelwlk 25778 The walks (in an undirected simple graph) are (ordered) pairs. (Contributed by Alexander van der Vekens, 30-Jun-2018.)
Rel (𝑉 Walks 𝐸)
 
Theoremwlks 25779* The set of walks (in an undirected graph). (Contributed by Alexander van der Vekens, 19-Oct-2017.)
((𝑉𝑋𝐸𝑌) → (𝑉 Walks 𝐸) = {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐸𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})})
 
Theoremiswlk 25780* Properties of a pair of functions to be a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → (𝐹(𝑉 Walks 𝐸)𝑃 ↔ (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
 
Theorem2mwlk 25781 The two mappings determining a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
(𝐹(𝑉 Walks 𝐸)𝑃 → (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉))
 
Theoremwlkres 25782* Restrictions of walks (i.e. special kinds of walks, for example paths - see pths 25828) are sets. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
(𝑓(𝑉𝑊𝐸)𝑝𝑓(𝑉 Walks 𝐸)𝑝)       ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝𝜑)} ∈ V)
 
Theoremwlkbprop 25783 Basic properties of a walk. (Contributed by Alexander van der Vekens, 31-Oct-2017.)
(𝐹(𝑉 Walks 𝐸)𝑃 → ((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
 
Theoremiswlkg 25784* Generalisation of iswlk 25780: Properties of a pair of functions to be a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 23-Jun-2018.)
((𝑉𝑋𝐸𝑌) → (𝐹(𝑉 Walks 𝐸)𝑃 ↔ (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
 
Theoremwlkcomp 25785* A walk expressed by properties of its components. (Contributed by Alexander van der Vekens, 23-Jun-2018.)
𝐹 = (1st𝑊)    &   𝑃 = (2nd𝑊)       ((𝑉𝑋𝐸𝑌𝑊 ∈ (𝑆 × 𝑇)) → (𝑊 ∈ (𝑉 Walks 𝐸) ↔ (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
 
Theoremwlkcompim 25786* Implications for the properties of the components of a walk. (Contributed by Alexander van der Vekens, 23-Jun-2018.)
𝐹 = (1st𝑊)    &   𝑃 = (2nd𝑊)       (𝑊 ∈ (𝑉 Walks 𝐸) → (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
 
Theoremwlkn0 25787 The set of vertices of a walk cannot be empty, i.e. a walk always consists of at least one vertex. (Contributed by Alexander van der Vekens, 19-Jul-2018.)
(𝐹(𝑉 Walks 𝐸)𝑃𝑃 ≠ ∅)
 
Theoremwlkop 25788 A walk (in an undirected simple graph) is an ordered pair. (Contributed by Alexander van der Vekens, 30-Jun-2018.)
(𝑊 ∈ (𝑉 Walks 𝐸) → 𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩)
 
Theoremwlkcpr 25789 A walk as class with two components. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
(𝑊 ∈ (𝑉 Walks 𝐸) ↔ (1st𝑊)(𝑉 Walks 𝐸)(2nd𝑊))
 
Theoremwlkelwrd 25790 The components of a walk are words/functions over a zero based range of integers. (Contributed by Alexander van der Vekens, 23-Jun-2018.)
(𝑊 ∈ (𝑉 Walks 𝐸) → ((1st𝑊) ∈ Word dom 𝐸 ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶𝑉))
 
Theoremedgwlk 25791* The (connected) edges of a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 22-Jul-2018.)
(𝐹(𝑉 Walks 𝐸)𝑃 → ∀𝑘 ∈ (0..^(#‘𝐹)){(𝑃𝑘), (𝑃‘(𝑘 + 1))} ∈ ran 𝐸)
 
Theoremwlklenvm1 25792 The number of edges of a walk (in an undirected graph) is the number of its vertices minus 1. (Contributed by Alexander van der Vekens, 1-Jul-2018.)
(𝐹(𝑉 Walks 𝐸)𝑃 → (#‘𝐹) = ((#‘𝑃) − 1))
 
Theoremwlkon 25793* The set of walks between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 12-Dec-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑉 WalkOn 𝐸)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵)})
 
Theoremiswlkon 25794 Properties of a pair of functions to be a walk between two given vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 2-Nov-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍) ∧ (𝐴𝑉𝐵𝑉)) → (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃 ↔ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)))
 
Theoremwlkonprop 25795 Properties of a walk between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
(𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃 → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)))
 
Theoremwlkoniswlk 25796 A walk between to vertices is a walk. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
(𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃𝐹(𝑉 Walks 𝐸)𝑃)
 
Theoremwlkonwlk 25797 A walk is a walk between its endpoints. (Contributed by Alexander van der Vekens, 2-Nov-2017.)
(𝐹(𝑉 Walks 𝐸)𝑃𝐹((𝑃‘0)(𝑉 WalkOn 𝐸)(𝑃‘(#‘𝐹)))𝑃)
 
Theoremtrls 25798* The set of trails (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
((𝑉𝑋𝐸𝑌) → (𝑉 Trails 𝐸) = {⟨𝑓, 𝑝⟩ ∣ ((𝑓 ∈ Word dom 𝐸 ∧ Fun 𝑓) ∧ 𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})})
 
Theoremistrl 25799* Properties of a pair of functions to be a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → (𝐹(𝑉 Trails 𝐸)𝑃 ↔ ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
 
Theoremistrl2 25800* Properties of a pair of functions to be a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → (𝐹(𝑉 Trails 𝐸)𝑃 ↔ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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